Throughout history it seems like there is an inconsistency and how the word empire is used. We call the Byzantine Empire, and Empire regardless of how small it shrank. Or even in the days of Aurelian when the Roman Empire split into four or five separate political units, we still refer to those individual territories as empires and not kingdoms.
So what really makes an Empire? Is a territorial size? Is it the amount of people there are subject to that particular political system? Why are there no empires today? Why do we not refer to the American Empire, or the Chinese Empire?
It feels like your question breaks into three parts: (1) the origin of the term, (2) usage history, and (3) current lack of use. Here is a very rough sketch and I encourage you to look into any of this that catches your eye more deeply - there is a lot to unpack.
Empire comes from the Roman term “imperium” which was the power to command - it’s what generals were given by the Roman people and Senate when they took up their command or what governors were given when they took up their posts. “Imperium” is related to “imperator” which originally referred to a military commander but later came to refer to the emperor after the rise of Augustus. Augustus and his successors had “imperium maius,” or greater imperium, which superseded that of other magistrates. Grants of this level of imperium - the sort needed to command multiple provinces or armies - were rarely given during the Roman Republic but became a hallmark of the reigning house in Rome. “Command” became synonymous with “the place that we command” and that’s where the Roman Empire (“Imperium Romanum”) came from.
Usage evolved from the Roman precedent. Addressing the specific cases you mention, we call the Byzantine Empire and, for example, the Gallic Empire empires because that’s what they called themselves, or at least their leaders called themselves emperors. The Byzantines also would have called themselves Romans but that name’s a complicated story in itself. In the European Middle Ages, Empire came to mean something larger than a kingdom as both a geographic and hierarchical distinction. Emperors ruled kings the way kings ruled dukes with the Holy Roman Empire (which was arguably not really holy, Roman, or an empire) as the classic example. All of this got wrapped up in European empire building schemes of the early modern and modern periods which saw the rise of the Russian, Spanish, British, French, etc. Empires. It drew on the vocabulary these peoples had inherited from the European Middle Ages but also harkened back to ideas of classical exceptionalism from Greece and Rome that were very much in vogue at the time and used as part of the justification for empire building. When the term is used for other historical entities, ex. the Hittite Empire, Aztec Empire, Mongol Empire, it is usually because that is how these entities have been categorized by scholars in the European tradition.
The 20th century saw empires fall apart, many after World War I, and the term went out of vogue as a political designation even if some countries, like the U.S. and China, arguably not only display traits associated with 19th century European empires but also maintain arguably imperial possessions, like Puerto Rico or Tibet.